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Division Algorithm for a Polynomial

If fx = 2x4...

Question

If $f (x) = 2 x^{4} - 5 x^{3} + x^{2} + 3 x - 2$ is divided by g(x) the qoutient is $q (x) = 2 x^{2} - 5 x + 3 a n d r (x) = - 2 x + 1$ find g(x).

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Solution

The division algorithm states that Dividend $=$ Divisor $\times$ Quotient $+$ Remainder that is $f (x) = g (x) \cdot q (x) + r (x)$

Here, it is given that the dividend is $f (x) = 2 x^{4} - 5 x^{3} + x^{2} + 3 x - 2$ , the divisor is $2 x^{2} - 5 x + 3$ and the remainder is $- 2 x + 1$ , therefore, by applying division algorithm we have:

$2 x^{4} - 5 x^{3} + x^{2} + 3 x - 2 = (2 x^{2} - 5 x + 3) g (x) + (- 2 x + 1) \Rightarrow 2 x^{4} - 5 x^{3} + x^{2} + 3 x - 2 - (- 2 x + 1) = (2 x^{2} - 5 x + 3) g (x) \Rightarrow 2 x^{4} - 5 x^{3} + x^{2} + 3 x - 2 + 2 x - 1 = (2 x^{2} - 5 x + 3) g (x) \Rightarrow 2 x^{4} - 5 x^{3} + x^{2} + 5 x - 3 = (2 x^{2} - 5 x + 3) g (x) \Rightarrow g (x) = \frac{2 x^{4} - 5 x^{3} + x^{2} + 5 x - 3}{2 x^{2} - 5 x + 3}$

Let us now divide $2 x^{4} - 5 x^{3} + x^{2} + 5 x - 3$ by $2 x^{2} - 5 x + 3$ as shown in the above image:

From the division, we observe that the quotient is $x^{2} - 1$ and the remainder is $0$ .

Hence, the quotient $q (x) = x^{2} - 1$ .

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